Inequalities

Inequalities

Overview

Inequalities is one of the most scoring topics in IBPS Clerk Prelims Reasoning section. The questions test your ability to understand relationships between elements (represented by letters like A, B, C) connected by mathematical symbols (>, <, =, ≥, ≤).

There are two types: Direct Inequalities where symbols are given as-is, and Coded Inequalities where symbols are replaced by codes (like @ means >, # means <). At Clerk level, the chains are shorter and the coding is simpler than PO exams. Master the symbol chain technique, and this becomes a guaranteed-marks topic.

The key skill is chain formation—combining multiple statements into one continuous chain to check if a conclusion is valid.

Key Concepts

• Basic symbols and their meaning: > (greater than), < (less than), = (equal to), ≥ (greater than or equal to), ≤ (less than or equal to).
• Chain compatibility rule: You can combine two statements only if they share a common element. Example: A > B and B ≥ C can combine to give A > B ≥ C.
• Direction rule: All inequality signs in a chain must point in the same direction (all left-pointing or all right-pointing) to derive a valid conclusion between the endpoints.
• Definite vs. Indefinite conclusions: A conclusion is definite only when you can trace a clear path with consistent direction. Mixed directions (like A > B < C) give no definite relation between A and C.
• Either-or condition: When two conclusions are individually false but together cover all possibilities (like A > B and A ≤ B), the answer is "either conclusion I or II follows."
• Priority of symbols: When combining ≥ and >, the result is >. When combining ≤ and <, the result is <. The "equal to" part is lost if any strict inequality exists in the chain.
• Complementary pairs: > and ≤ are complements. < and ≥ are complements. If one is false, the other must be true.

Formulas / Key Facts

Quick symbol priority: Strict inequality (>, <) dominates over non-strict (≥, ≤) when combined.

Coded inequality step: Always decode first, then solve as direct inequality.

Worked Examples

Example 1: Direct Inequality

Statement: P ≥ Q > R = S ≤ T

Conclusions: I. P > R II. T ≥ R

Solution:

• For I: Trace P to R → P ≥ Q > R. Direction is consistent (left to right, all showing P is greater side). Combining ≥ and > gives >. So P > R. Conclusion I follows.
• For II: Trace T to R → T ≥ S = R. Reading from T: T ≥ S = R means T ≥ R. Conclusion II follows.

Answer: Both I and II follow.

---

Example 2: Coded Inequality

Code: @ means >, # means <, $ means =, % means ≥, & means ≤

Statement: A % B @ C # D $ E

Decode first: A ≥ B > C < D = E

Conclusions: I. A > C II. E > C

Solution:

• For I: A ≥ B > C. Consistent direction. ≥ combined with > gives >. So A > C. Follows.
• For II: E = D > C (reading right to left, D > C means D is greater than C, and E = D). So E > C. Follows.

Answer: Both follow.

---

Example 3: Either-Or Case

Statement: M > N ≤ O = P

Conclusions: I. M > P II. M ≤ P

Solution:

• Trace M to P: M > N ≤ O = P. Direction changes at N (> then ≤). No definite relation between M and P.
• Conclusion I (M > P): Not definite. Does not follow.
• Conclusion II (M ≤ P): Not definite. Does not follow.
• But M > P and M ≤ P are complementary (one must be true). Either I or II follows.

Answer: Either I or II follows.

Common Mistakes

• Ignoring direction change: Students trace A > B < C and conclude A > C. Fix: When signs point opposite directions, no relation exists between endpoints.
• Confusing ≥ with >: Writing "A is greater than B" when the symbol is ≥. Fix: ≥ means "greater than OR equal to"—both possibilities exist.
• Forgetting to decode in coded inequalities: Jumping to solve without converting symbols. Fix: Always write the decoded statement first, then trace.
• Missing the either-or case: Marking "neither follows" when two complementary conclusions are given. Fix: Check if conclusions are exact complements (like > and ≤). If yes, either-or applies.
• Reversing the chain incorrectly: If A > B, students sometimes read B > A when tracing backward. Fix: A > B is the same as B < A. The smaller element stays smaller.

Quick Reference

• Same direction = definite conclusion; mixed direction = no conclusion.
• Strict symbol (>, <) + any symbol = strict symbol in result.
• Either-or applies only when conclusions are exact complements.
• Always decode first in coded inequalities—never solve with symbols.
• For ≥ or ≤, the "equal" possibility means you cannot claim strict inequality.
• Trace element to element; the chain must be continuous with no gaps.